A new class of almost complex structures
type: Item
instance of: scholarly article
title: A new class of almost complex structures
zbMATH Open document ID: 0862.53051
DOI: 10.1007/BF01759257
author: Chuan-Chih Hsiung
author: Bonnie Xiong
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 1997-05-25T00:00:00Z
review text: An almost Hermitian manifold \((M,g,J)\) such that \[ R(JX,JY,Z,W) + R(JY, JZ, X,W) + R(JZ,JX,Y,W)=0, \] where \(R\) denotes the Riemannian curvature of \(g\), is called almost Hermitian \(C\)-manifold. The main results of this paper are: (1) an analogue of the Schur lemma for the holomorphic sectional curvature of an almost Hermitian \(C\)-manifold is proved; and (2) the following theorem is proved: ``Let \((M,g,J)\) be an almost Hermitian \(C\)-manifold with nonzero pointwise constant holomorphic sectional curvature such that \[ \sum_i \biggl\{R \bigl(X,Y, (\nabla_ZJ) e_i,e_i \bigr) + R \bigl(Y,Z, (\nabla_XJ) e_i,e _i \bigr) + R \bigl(Z, X, (\nabla_YJ) e_i,e_i \bigr) \biggr\} = 0\tag{*} \] is satisfied (here \(\nabla\) is the Levi-Civita connection and \(\{e_i\}\) is a local orthonormal frame on \(M)\). Then \((M,g,J)\) is Kählerian.'' No example of an almost Hermitian non-Kähler manifold which satisfies (*) is given.
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 53C15
zbMATH DE Number: 878445
zbMATH Keywords: almost Hermitian \(C\)-manifold
reviewed by: Luis A. Cordero
MaRDI profile type: Publication
cites work: Q5837114
cites work: The sixteen classes of almost Hermitian manifolds and their linear invariants
cites work: Q3746457
cites work: Q4385252
cites work: Q5749911
cites work: Q5815997
cites work: Varieta parakähleriane
cites work: Almost Hermitian manifolds with constant holomorphic sectional curvature
cites work: Structure theorems on Riemannian spaces satisfying \(R(X,Y)\cdot R=0\). I: The local version
cites work: Q4112572
cites work: Curvature and Betti Numbers. (AM-32)
description: scientific article; zbMATH DE number 878445